1 Definition

The main feature of a probability distribution function is that it induces a probability measureP on the measure space (Ω,𝔅), given by

P⁢(A):=∫Af⁢(x)⁢𝑑μ=∫Ω1A⁢f⁢(x)⁢𝑑μ,

for all A∈𝔅. The measureP is called the associated probability measure of f. Note that P and μ are different measures, though they both share the same underlying measurable space(Ω,𝔅).

2 Examples

2.1 Discrete case

Let I be a countable set, and impose the counting measure on I (μ⁢(A):=|A|, the cardinality of A, for any subset A⊂I). A probability distribution function on I is then a nonnegative function f:I⟶ℝ satisfying the equation

2.2 Continuous case

Suppose (Ω,𝔅,μ) equals (ℝ,𝔅λ,λ), the real numbers equipped with Lebesgue measure. Then a probability distribution function f:ℝ⟶ℝ is simply a measurable, nonnegative almost everywhere function such that

∫-∞∞f⁢(x)⁢𝑑x=1.

The associated measure has Radon–Nikodym derivative (http://planetmath.org/RadonNikodymTheorem) with respect to λ equal to f: